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A002295
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Number of dissections of a polygon: C(6n,n)/(5n+1).
(Formerly M4260 N1780)
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12
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1, 1, 6, 51, 506, 5481, 62832, 749398, 9203634, 115607310, 1478314266, 19180049928, 251857119696, 3340843549855, 44700485049720, 602574657427116, 8175951659117794, 111572030260242090, 1530312970340384580, 21085148778264281865, 291705220704719165526
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n), n>=1, enumerates sextic (6-ary) trees (rooted, ordered, incomplete) with n vertices (including the root).
Pfaff-Fuss-Catalan sequence C^{m}_n for m=6. See the Graham et al. reference, p. 347. eq. 7.66. See also the P\'olya-Szeg\"o reference.
Also 6-Raney sequence. See the Graham et al. reference, p. 346-7.
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REFERENCES
| R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pp. 200, 347.
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
G. P\'olya and G. Szeg\"o, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem 211, p. 146 with solution on p. 348.
Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (5).
Editor's note: "Ueber die Bestimmung der Anzahl der verschiedenen Arten, auf welche sich ein n-Eck durch Diagonalen in lauter m-Ecke zerlegen laesst, mit Bezug auf einige Abhandlungen der Herren Lame, Rodrigues, Binet, Catalan und Duhamel in dem Journal de Mathematiques pure et appliquees, publie par Joseph Liouville. T. III. IV.", Archiv der Mathematik u. Physik, 1 (1841), pp. 193ff; see especially p. 198.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 288
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FORMULA
| O.g.f. A(x)= 1 + x*A(x)^6 = 1/(1-x*A(x)^5).
a(n)=binomial(6*n,n-1)/n, n>=1, a(0)=1. From the Lagrange series of the o.g.f. A(x) with its above given implicit equation.
a(n) = upper left term in M^n, M = the production matrix:
1, 1
5, 5, 1
15, 15, 5, 1
35, 35, 15, 5, 1
...
(where (1, 5, 15, 35,...) = A000332 starting with 1. - Gary W. Adamson, Jul 08 2011
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EXAMPLE
| There are a(2)=6 sextic trees (vertex degree <=6 and 6 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these 6 trees yields 6*6+binomial(6,2)=51=a(3) such trees.
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MATHEMATICA
| Table[Binomial[6n, n]/(5n + 1), {n, 0, 20}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 06 2006
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CROSSREFS
| Cf. A002294, A002296.
Fifth column of triangle A062993.
Sequence in context: A202754 A180901 A199685 * A027393 A124565 A057817
Adjacent sequences: A002292 A002293 A002294 * A002296 A002297 A002298
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KEYWORD
| easy,nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 06 2006
Pfaff-Fuss-Catalan, Raney and 6-ary tree comments from W. Lang, Sep 14 2007.
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